Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An acceleration technique for methods for finding the nearest point in a polytope and computing the distance between two polytopes

M. V. Dolgopolik

TL;DR

A simple and efficient acceleration technique for an arbitrary method for computing the Euclidean projection of a point onto a convex polytope, defined as the convex hull of a finite number of points, in the case when the number of touches is much greater than the dimension of the space.

Abstract

We present a simple and efficient acceleration technique for an arbitrary method for computing the Euclidean projection of a point onto a convex polytope, defined as the convex hull of a finite number of points, in the case when the number of points in the polytope is much greater than the dimension of the space. The technique consists in applying any given method to a "small" subpolytope of the original polytope and gradually shifting it, till the projection of the given point onto the subpolytope coincides with its projection onto the original polytope. The results of numerical experiments demonstrate the high efficiency of the proposed acceleration technique. In particular, they show that the reduction of computation time increases with an increase of the number of points in the polytope and is proportional to this number for some methods. In the second part of the paper, we also discuss a straightforward extension of the proposed acceleration technique to the case of arbitrary methods for computing the distance between two convex polytopes, defined as the convex hulls of finite sets of points.

An acceleration technique for methods for finding the nearest point in a polytope and computing the distance between two polytopes

TL;DR

A simple and efficient acceleration technique for an arbitrary method for computing the Euclidean projection of a point onto a convex polytope, defined as the convex hull of a finite number of points, in the case when the number of touches is much greater than the dimension of the space.

Abstract

We present a simple and efficient acceleration technique for an arbitrary method for computing the Euclidean projection of a point onto a convex polytope, defined as the convex hull of a finite number of points, in the case when the number of points in the polytope is much greater than the dimension of the space. The technique consists in applying any given method to a "small" subpolytope of the original polytope and gradually shifting it, till the projection of the given point onto the subpolytope coincides with its projection onto the original polytope. The results of numerical experiments demonstrate the high efficiency of the proposed acceleration technique. In particular, they show that the reduction of computation time increases with an increase of the number of points in the polytope and is proportional to this number for some methods. In the second part of the paper, we also discuss a straightforward extension of the proposed acceleration technique to the case of arbitrary methods for computing the distance between two convex polytopes, defined as the convex hulls of finite sets of points.
Paper Structure (11 sections, 20 theorems, 134 equations, 6 figures, 4 algorithms)

This paper contains 11 sections, 20 theorems, 134 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Finding the nearest point in a polytope
  3. General acceleration technique
  4. The steepest descent exchange rule
  5. A robust version of the meta-algorithm
  6. Numerical experiments
  7. A comparison with the Wolfe method
  8. Computing the distance between two polytopes
  9. An extension of the acceleration technique
  10. A robust version of the meta-algorithm
  11. Numerical experiments

Key Result

Proposition 1

A point $x_* \in P$ is a globally optimal solution of the problem $(\mathcal{P})$ if and only if

Figures (6)

  • Figure 1: The results of numerical experiments in the case $d = 3$ for quadprog routine (left figure) and the MDM method (right figure).
  • Figure 2: The results of numerical experiments in the case $d = 10$ for quadprog routine (left figure) and the MDM method (right figure).
  • Figure 3: The results of numerical experiments in the case $d = 50$ for quadprog routine (left figure) and the Wolfe method (right figure).
  • Figure 4: The average number of outer loops (iterations/shifts of the subpolytope) for Meta-algorithm \ref{['alg:NearestPointRobust']} and the Wolfe method in the case $d = 50$.
  • Figure 5: The results of numerical experiments in the case $d = 3$ for quadprog routine (left figure) and the ALT-MDM method (right figure).
  • ...and 1 more figures

Theorems & Definitions (48)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Lemma 2
  • proof
  • Remark 4
  • Theorem 1
  • ...and 38 more